Time-frequency Localization Properties Research Articles

Pseudopolar Format Matrix Description of Near-Range Radar Imaging and Fractional Fourier Transform

Near-range radar imaging (NRRI) has evolved into a vital technology with diverse applications spanning fields such as remote sensing, surveillance, medical imaging and non-destructive testing. The Pseudopolar Format Matrix (PFM) has emerged as a promising technique for representing radar data in a compact and efficient manner. In this paper, we present a comprehensive PFM description of near-range radar imaging. Furthermore, this paper also explores the integration of the Fractional Fourier Transform (FrFT) with PFM for enhanced radar signal analysis. The FrFT—a powerful mathematical tool for signal processing—offers unique capabilities in analysing signals with time-frequency localization properties. By combining FrFT with PFM, we have achieved significant advancements in radar imaging, particularly in dealing with complex clutter environments and improving target detection accuracy. Meanwhile, this paper highlights the imaging matrix form of FrFT under the PFM, emphasizing the potential for addressing challenges encountered in near-range radar imaging. Finally, numerical simulation and real-world scenario measurement imaging results verify optimized accuracy and computational efficiency with the fusion of PFM and FrFT techniques, paving the way for further innovations in near-range radar imaging applications.

An efficient deep learning framework for P300 evoked related potential detection in EEG signal

BackgroundIncorporating the time-frequency localization properties of Gabor transform (GT), the complexity understandings of convolutional neural network (CNN), and histogram of oriented gradients (HOG) efficacy in distinguishing positive peaks can exhibit their characteristics to reveal an effective solution in the detection of P300 evoked related potential (ERP). By applying a drastic number of convolutional layers, the majority of deep networks elicit sufficient properties for the output determination, leading to gigantic and time-consuming structures. In this paper, we propose a novel deep learning framework by the combination of tuned GT, and modified HOG with the CNN as “TGT-MHOG-CNN” for detection of P300 ERP in EEG signal. MethodIn the proposed method, GT is tuned based on triangular function for EEG signals, and then spectrograms including time-frequency information are captured. The function's parameters are justified to differentiate the signals with the P300 component. Furthermore, HOG is modified (MHOG) for the 2-D EEG signal, and consequently, gradients patterns are extracted for the target potentials. MHOG is potent in distinguishing the positive peak in the general waveform; however, GT unravels time-frequency information, which is ignored in the gradient histogram. These outputs of GT and MHOG do not share the same nature in the images nor overlap. Therefore, more extensive information is reached without redundancy or excessive information by fusing them. Combining GT and MHOG provides different patterns which benefit CNN for more precise detection. Consequently, TGT-MHOG-CNN ends in a more straightforward structure than other networks, and therefore, the whole performance is acceptable with faster rates and very high accuracy. ResultsBCI Competition II and III datasets are used to evaluate the performance of the proposed method. These datasets include a complete record for P300 ERP with BCI2000 using a paradigm, and it has numerous noises, including power and muscle-based noises. The objective is to predict the correct character in each provided character selection epochs. Compared to state-of-the-art methods, simulation results indicate striking abilities of the proposed framework for P300 ERP detection. Our best record reached the P300 ERP classification rates of over 98.7% accuracy and 98.7% precision for BCI Competition II and 99% accuracy and 100% precision for BCI Competition III datasets, with superiority in execution time for the mentioned datasets.

Efficient Estimation of Empirical Green’s Function by Removing Transients From Sensor Data Using Time–Frequency Normalization Technique

Clean ambient noise data is the preliminary requirement to perform ambient noise tomography of the earth. Conventional 1-bit time-domain normalization (TDN) is the most popular method to remove the effect of earthquakes and other transient signals obscured by the continuous recording of ambient noise at the sensors. The presence of monochromatic noise sources is also reduced by performing spectral whitening. In this manner, recorded ambient noise data is first normalized in the time domain and then in the frequency domain. It is observed that these normalization procedures do not make the amplitude “even” within the small segments of the complete normalized ambient noise time series and hence, ambient noise remains nondiffusive within these segments. Therefore, in this article, a time–frequency normalization technique is proposed to efficiently remove the temporal and spectral components of the unwanted signals. In this article, continuous wavelet transform (CWT) is used to transform raw ambient noise time series to time–frequency domain to exploit its time–frequency localization property. Complex Morlet wavelet best captures the oscillations present in the raw ambient noise data and then upon normalizing CWT coefficients, a nearly diffusive ambient noise data is retrieved which is suitable to estimate empirical Green’s function (EGF). The performance of the proposed method is tested using synthetic data and the EGFs are estimated using real ambient noise data of a subset of STS-2 USArray sensors. The EGFs are evaluated quantitatively by calculating the signal-to-noise ratio (SNR) and comparing them with those estimated using the conventional method.

Digital Image Watermarking Technique Using Discrete Wavelet Transform and Discrete Cosine Transform

Digital watermarking has been proposed as a viable solution to the need of copyright protection and authentication of multimedia data in a networked environment, since it makes possible to identify the author, owner, distributor or authorized consumer of a document. In this paper a new watermarking technique to add a code to digital images is presented: the method operates in the frequency domain embedding a pseudo-random sequence of real numbers in a selected set of DCT coefficient. And a new method for digital image watermarking which does not require the original image for watermark detection. The watermark is added in select coefficients with significant image energy in the transform domain in order to ensure non- erasability of the watermark. Advantages of the proposed method include: improved resistance to attacks on the watermark, implicit visual masking utilizing the time-frequency localization property of wavelet transform and a robust definition for the threshold which validates the watermark.. Experimental results demonstrate that this proposed technique is robust to most of the signal processing techniques and geometric distortions.

Analysis of the Time-Frequency Localization Property of the Filter Banks for Optical OFDM/OQAM Systems

Orthogonal frequency-division multiplexing offset-quadrature amplitude modulation (OFDM/OQAM) relaxes the orthogonal condition from the complex domain to the real field. Therefore, the cyclic prefix (CP) inserted between consecutive optical OFDM blocks, for combating the inter-symbol interference (ISI) induced by chromatic dispersion (CD) and polarization-mode-dispersion (PMD), could be removed for optical OFDM/OQAM to promote the system spectral efficiency. The ISI and the inter-carrier-interference (ICI) could be combated by using the filter banks with good time-frequency localization (TFL) property. Thus, the accuracy of channel estimation and system transmission performance largely depend on the TFL property of the filter banks for optical OFDM/OQAM. Recently, numerical experimental demonstrations and theoretical discussions of optical OFDM/OQAM have been reported. However, to our best knowledge, discussions on the TFL property of the filter banks have not been studied for optical OFDM/OQAM to date. In this paper, we systematically analyze the TFL property of the filter banks for optical OFDM/OQAM for the first time. We evaluate the TFL property of the filter banks with energy concentration, orthogonal basis, and isotropic property. Three typical filters as the Gaussian filter, the extended Gaussian function (EGF) filter and the isotropic orthogonal transfer algorithm (IOTA) filter have been employed for comparison. According to our theoretical analysis, with the use of the IOTA filter, optical impairments and channel estimation complexities could be decreased evidently thanks to the promising isotropic property. As shown in the numerous Montel Carlo simulations of the coherent optical OFDM/OQAM, system robustness against IMI, phase noise induced interference (PNII) and nonlinear effect could be improved evidently with the use of the IOTA filter.

Doubly Orthogonal Wavelet Packets for Multi-Users Indoor Visible Light Communication Systems

Visible Light Communication (VLC) is a data communication technology that modulates the intensity of the light to transmit the information mostly by means of Light Emitting Diodes (LEDs). The data rate is mainly throttled by the limited bandwidth of the LEDs. To combat, Multi-carrier Code Division Multiple Access (MC-CDMA) is a favorable technique for achieving higher data rates along with reduced Inter-Symbol Interference (ISI) and easy access to multi-users at the cost of slightly reduced compromised spectral efficiency and Multiple Access Interference (MAI). In this article, a multi-user VLC system is designed using a Discrete Wavelet Transform (DWT) that eradicates the use of cyclic prefix due to the good orthogonality and time-frequency localization properties of wavelets. Moreover, the design also comprises suitable signature codes, which are generated by employing double orthogonality depending upon Walsh codes and Wavelet Packets. The proposed multi-user system is simulated in MATLAB software and its overall performance is assessed using line-of-sight (LoS) and non-line-of-sight (NLoS) configurations. Furthermore, two sub-optimum multi-users detection schemes such as zero forcing (ZF) and minimum-mean-square-error (MMSE) are also used at the receiver. The simulated results illustrate that the doubly orthogonal signature waveform-based DWT-MC-CDMA with MMSE detection scheme outperforms the Walsh code-based multi-user system.

Fuzzy wavelet extreme learning machine

Incorporating the time-frequency localization properties of wavelets and the learning abilities of neural network (NN), the approximate reasoning characteristics of fuzzy inference system and the advantages of ELM (one-pass learning and good generalization performance at extremely fast learning speed) can exhibit their characteristics to reveal an effective solution in many applications. Following that, this paper presents a novel structure called fuzzy wavelet extreme learning machine (FW-ELM). The main objectives of FW-ELM are to significantly reduce the network complexity by reducing the number of linear learning parameters, and to decrease the sensitivity to random initialization procedure while the acceptable accuracy and generalization performances are preserved. In the proposed structure, each fuzzy rule corresponds to a sub-wavelet neural network and consists of wavelets with different dilations and translations. In this model, in order to achieve a balance between network complexity and performance accuracy, in the THEN-part of each fuzzy rule, one coefficient is considered for each two inputs. In this work, first, the equivalence of an FW model and an SLFN is proved and then ELM can be directly applied to the model. All free parameters of membership functions and wavelet coefficients are generated randomly, and only the output weights are determined analytically using a one-pass learning method. To evaluate FW-ELM, it is compared with popular fuzzy models like OS-Fuzzy-ELM, Simpl_eTS, ANFIS and several other relevant algorithms such as ELM, BP and SVR on various benchmark datasets. Simulation results demonstrate the remarkable efficiency resulting from the proposed approach. Performance accuracy of FW-ELM is shown to be comparable with OS-Fuzzy-ELM and better than the rest of the well-known methods.

Pulse shaping design for OFDM systems

Spectrally contained OFDM-based waveforms are considered key enablers for a flexible air interface design to support a broad range of services and frequencies as envisaged for 5G mobile systems. By allowing for the flexible configuration of physical layer parameters in response to diverse requirements, these waveforms enable the in-band coexistence of different services. One candidate from this category of waveforms is pulse-shaped OFDM, which follows the idea of subcarrier filtering while fully maintaining the compatibility with CP-OFDM. In this paper, we provide an overview of pulse shaping methods in OFDM systems and propose a new pulse-shaped design method with arbitrary length constraint and good time-frequency localization property. Based on the pulse design, we discuss different receiver realizations and present a criterion for pulse shape evaluation. In addition, the parameterizations of OFDM system to address diverse requirements of the services envisaged for the 5G systems are described. Link and system performance results for selected scenarios show that a proper design of the OFDM numerologies and pulse shapes could substantially improve the performance under time and frequency distortions. Furthermore, pulse-shaped OFDM is able to support asynchronous transmissions and reduce the signal sensitivity to Doppler distortions, rendering it beneficial for various applications from the context of vehicular communications and the Internet-of-things.

A noise reduction method for Ground Penetrating Radar signal based on wavelet transform and application in tunnel lining

One of the major limitations that hinder the use of ground penetrating radar (GPR) in civil engineering is the inevitable noise interference associated with the GPR signal. This study proposes a noise reduction technology for a GPR signal by utilizing the excellent time-frequency localization properties of a wavelet and by analysing the characteristics of the wavelet function and the principles of wavelet noise reduction. A Daubechies wavelet is selected as the wavelet function, and the wavelet decomposition and re-construction is performed on a single channel radar signal. The images of tunnel lining from GPR are compared before and after wavelet transform, and the wavelet transform processing is proved to be capable of separating noise as well as of improving the signal-to-noise ratio. The results show that wavelet transform processing is found useful to improve the recognition of image and the accuracy of radar signal interpretation.

Discrete framelet transform based OFDM system using N-FRAT mapping over Rayleigh fading channel

This article proposes an N-finite Radon transform (N-FRAT) and a discrete framelet transform (FT) for the design of a new orthogonal frequency division multiplexing (OFDM) system. In the proposed system, the inverse FT (IFT) and FT were used instead of the inverse fast Fourier transform (IFFT) and fast Fourier transform (FFT) to realize multicarrier modulation and demodulation techniques, respectively. Simultaneously, it was suggested that the N-FRAT technique operate as a data mapper instead of the conventional phase shift keying (PSK) and quadrature amplitude mapping (QAM), which are usually used with the traditional OFDM system. The usage of the N-FRAT and FT in the proposed system led to an increase in the orthogonality between the subcarriers due to the application of IFFT in the N-FRAT, which was added to improve the orthogonality and time–frequency localization properties of the FT. Furthermore the proposed structure improved the bandwidth efficiency through the elimination of the cyclic prefix compared to the traditional system. The proposed system was simulated and compared with other OFDM systems, such as FFT based OFDM using N-FRAT mapping (N-FRAT-FFT-OFDM), FT based OFDM using QAM mapping (QAM-FT-OFDM), and FFT based OFDM using QAM mapping (QAM-FFT-OFDM). The simulation was performed over frequency selective fading channel using MATLAB technical programming language. The results of simulations showed that the new structure outperformed the other three systems by reducing the inter-symbol interference (ISI) and inter-carrier interference (ICI), which then improved the bit error rate (BER) performance.

Discrete Multiwavelet Critical-Sampling Transform-Based OFDM System over Rayleigh Fading Channels

Discrete multiwavelet critical-sampling transform (DMWCST) has been proposed instead of fast Fourier transform (FFT) in the realization of the orthogonal frequency division multiplexing (OFDM) system. The proposed structure further reduces the level of interference and improves the bandwidth efficiency through the elimination of the cyclic prefix due to the good orthogonality and time-frequency localization properties of the multiwavelet transform. The proposed system was simulated using MATLAB to allow various parameters of the system to be varied and tested. The performance of DMWCST-based OFDM (DMWCST-OFDM) was compared with that of the discrete wavelet transform-based OFDM (DWT-OFDM) and the traditional FFT-based OFDM (FFT-OFDM) over flat fading and frequency-selective fading channels. Results obtained indicate that the performance of the proposed DMWCST-OFDM system achieves significant improvement compared to those of DWT-OFDM and FFT-OFDM systems. DMWCST improves the performance of the OFDM system by a factor of 1.5–2.5 dB and 13–15.5 dB compared with the DWT and FFT, respectively. Therefore the proposed system offers higher data rate in wireless mobile communications.

Coordinated Adaptive Control of Multiple Flexible AC Transmission Systems using Multiple-input—Multiple-output Neuro-fuzzy Damping Control Paradigms

This article presents a novel online adaptive non-linear multiple-input–multiple-output neuro-fuzzy control scheme for flexible AC transmission systems. The proposed control system synergistically integrates the time-frequency localization property of wavelets with locally controllable membership functions in the structure of neuro-fuzzy Takagi–Sugeno–Kang control. The proposed control scheme is checked for both multiple-input–single-output and multiple-input–multiple-output topologies. The performance of the proposed control system is validated against different contingencies and operating conditions using non-linear time-domain simulations and different performance indices. The results reveal that the proposed control scheme effectively damps the local and inter-area modes of oscillations.

Research of the Transient Signal Recognition Based on Empirical Mode Decomposition

This paper introduces the empirical mode decomposition (EMD) method of the basic theory, problems and means to solve. Apply the approach to mechanical vibration signal containing a transient pulse processing and analysis carried out, and the wavelet time-frequency analysis methods are compared, the results show that it can effectively decompose nonlinear and non-stationary vibration signals, and has a self-adaptive, and in the time domain and frequency domain have better resolution capabilities, and the component with a more clear physical meaning. Due to its diversity of showing the results, you can make further precise analysis of a single component, and the transient signals can be effectively recognized, and can locate mutation point in time, describing the time-frequency localization properties. EMD, transient signals, mechanical vibration

Method of Harmonic Detection Based on the Wavelet Transform

The harmonic current detection technology is one of the key technologies of active power filter technologies. The development of the harmonic current detection technology directly determines the development of the active power filter technologies. Based on this, this paper introduces some basis concepts of wavelet transform and analyzes its time-frequency localization properties, then, describes the harmonic detection methods based on wavelet transform in terms of program building, algorithm selection and wavelet function selection. The results show that the harmonic current detection methods based on wavelet transform are able to compensate the inadequacy of Fourier transforms and can achieve the functions of detecting the steady-state and time-varying harmonic current of the grid in harmonic detection of active power filter.

Relevance of Wavelet Shape Selection in a complex signal

In wavelet analysis, the signal reconstruction and statistical estimators are strongly influenced by the wavelet shape that controls the time-frequency localization properties. However, this dependency does not imply that the wavelet form is physically relevant to extract spatially invariant wavelet-based shape signatures. A new statistical estimator is proposed in order to quantify the influence of the wavelet on the spatial detection. This indicator is then used to analyze signals obtained from the topography of abraded surfaces. It is shown that the Coiflet wavelet is physically adapted to reproduce the elementary mechanical process which creates the abrasion of the surface. Using 8 different wavelets, the analysis of the signal obtained by scanning the abraded surface leads to the same spatial localization regardless the parameters of the abrasive process and whatever the wavelet shapes. No statistical difference related to the type of wavelets is found between the indicators (RMS, spectral moments…) extracted from the reconstructed signals calculated on different scales and for various abrasive processes. If the wavelet decomposition is seen as a multiscale microscope, a surface can be seen in different ways according to the type of wavelets. However, the morphological changes of the surface caused by external mechanical causes and characterized by several statistical parameters are statistically similar regardless the shape of the wavelets.

Voltage Flicker Extraction Based on Wavelet Analysis

Extracting voltage flicker from the sampling voltage signal is a precondition for management of flicker. Voltage flicker signal is a low frequency time-varying non-stationary signal. The traditional fourier transform has great limitations when analyze the non-stationary signal for not having the time resolution. As wavelet transform has good property of time-frequency localization, it become a powerful tool for analyze this kind of signal. This paper adopts multi-resolution analysis of wavelet to extract voltage flicker signal. Furthermore, according to the characteristics of wavelet function, this paper selects Daubechies wavelet to accomplish the multi-level decomposition and reconstruction of signal, in order to get the frequency and amplitude of voltage flicker signals. Based on the principle of modulus maximum, it can be determined what time the voltage flicker happen and over. The results of MATLAB simulation indicate that voltage flicker signal can be effectively extracted by wavelet multi-resolution analysis. Wavelet multi-resolution analysis is considerably ideal for voltage flicker extraction.

Fractional Hilbert transform extensions and associated analytic signal construction

The analytic signal (AS) was proposed by Gabor as a complex signal corresponding to a given real signal. The AS has a one-sided spectrum and gives rise to meaningful spectral averages. The Hilbert transform (HT) is a key component in Gabor's AS construction. We generalize the construction methodology by employing the fractional Hilbert transform (FrHT), without going through the standard fractional Fourier transform (FrFT) route. We discuss some properties of the fractional Hilbert operator and show how decomposition of the operator in terms of the identity and the standard Hilbert operators enables the construction of a family of analytic signals. We show that these analytic signals also satisfy Bedrosian-type properties and that their time–frequency localization properties are unaltered. We also propose a generalized-phase AS (GPAS) using a generalized-phase Hilbert transform (GPHT). We show that the GPHT shares many properties of the FrHT, in particular, selective highlighting of singularities, and a connection with Lie groups. We also investigate the duality between analyticity and causality concepts to arrive at a representation of causal signals in terms of the FrHT and GPHT. On the application front, we develop a secure multi-key single-sideband (SSB) modulation scheme and analyze its performance in noise and sensitivity to security key perturbations.

Bandlimited shearlet-type frames with nice duals

The present paper constructs a frame/dual frame pair of shearlet type such that both frames possess the distinctive time–frequency localization properties needed in establishing their desirable approximation properties. Our construction is based on a careful pasting together of two bandlimited shearlet Parseval frames associated with two different frequency cones, inspired by domain decomposition methods used primarily for the solution of PDEs.

High-Power AC Servo System Identification Research Based on Wavelet Neural Network

As a result of the non-linear characteristics and the uncertain disturbances in high-power AC servo system, it is difficult to construct an accurate mathematical model. In order to solve this problem, this article proposes a system identification method based on wavelet neural network. It makes full use of the advantages of the wavelet which combines neural network good time-frequency localization property and volatility of wavelet function and the nonlinear mapping capacity, self-learning and adaptive capacity of neural networks to solve the problem of non-unique RBF neural network approximation function expression. The simulation results show that the convergence rate, robustness and approximation accuracy of this method are better than the traditional neural network.

Wavelet-based feature extraction using probabilistic finite state automata for pattern classification

Real-time data-driven pattern classification requires extraction of relevant features from the observed time series as low-dimensional and yet information-rich representations of the underlying dynamics. These low-dimensional features facilitate in situ decision-making in diverse applications, such as computer vision, structural health monitoring, and robotics. Wavelet transforms of time series have been widely used for feature extraction owing to their time-frequency localization properties. In this regard, this paper presents a symbolic dynamics-based method to model surface images, generated by wavelet coefficients in the scale-shift space. These symbolic dynamics-based models (e.g., probabilistic finite state automata (PFSA)) capture the relevant information, embedded in the sensor data, from the associated Perron-Frobenius operators (i.e., the state-transition probability matrices). The proposed method of pattern classification has been experimentally validated on laboratory apparatuses for two different applications: (i) early detection of evolving damage in polycrystalline alloy structures, and (ii) classification of mobile robots and their motion profiles.